Abstract
We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.
Highlights
We construct a divided difference operator using GKM theory
A classical problem of Schubert calculus is to define explicit classes S[w] to represent Schubert varieties in cohomology rings of a partial flag variety. For geometric reasons these classes form an additive basis for the cohomology. In equivariant cohomology this problem reduces to finding the polynomials S[w]([v]) which are nonzero only if [v] ≥ [w] in Bruhat order
For more general spaces the uniqueness or even existence of generalized Schubert classes named flow-up classes is not guaranteed. When they exist it is natural to ask for some combinatorial formula defining the polynomials
Summary
25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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